Abstract

We studied both theoretically and experimentally the intensity distribution of a Gaussian laser beam when it was focused by an objective lens with its numerical aperture up to 0.95. Approximate formulas for full width at half-maximum (FWHM) of the intensity distribution at focus were derived for very large and very small initial beam waists with respect to the entrance pupil radius of the objective lens. In experiments, the energy flux through a 0.5μm pinhole was measured for various pinhole positions. It was found in theoretical analysis and confirmed in experiments that the FWHMs at focus in the transverse and longitudinal directions do not increase much from the ultimate FWHMs until the input beam waist is reduced below half of the entrance pupil radius.

© 2006 Optical Society of America

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References

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  1. G. F. Marshall, ed., Optical Scanning (Marcel Dekker, 1991).
  2. P. Belland and J. P. Crenn, 'Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,' Appl. Opt. 21, 522-527 (1982).
    [CrossRef] [PubMed]
  3. K. Tanaka, N. Saga, and K. Hauchi, 'Focusing of a Gaussian beam through a finite aperture lens,' Appl. Opt. 24, 1382-1390 (1985).
    [CrossRef]
  4. E. Wolf, 'Electromagnetic diffraction in optical systems. I. An integral representation of the image field,' Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
    [CrossRef]
  5. B. Richards and E. Wolf, 'Electromagnetic diffraction in optical systems. Il. Structure of the image field in an aplanatic system,' Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]
  6. M. Mansuripur, 'Certain computational aspects of vector diffraction problems,' J. Opt. Soc. Am. A 6, 786-805 (1989).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (expanded) (Cambridge U. Press, 1999).
  8. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).
  9. M. B. Schneider and W. W. Webb, 'Measurement of submicron laser beam radii,' Appl. Opt. 20, 1382-1388 (1981).
    [CrossRef] [PubMed]
  10. A. H. Firester, M. E. Heller, and P. Sheng, 'Knife-edge scanning measurements of subwavelength focused light beams,' Appl. Opt. 16, 1971-1974 (1977).
    [CrossRef] [PubMed]
  11. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
    [CrossRef]
  12. R. Dorn, S. Quabis, and G. Leuchs, 'The focus of light--linear polarization breaks the rotational symmetry of the focal spot,' J. Mod. Opt. 50, 1917-1926 (2003).
    [CrossRef]
  13. S. K. Rhodes, A. Barty, A. Roberts, and K. A. Nugent, 'Sub-wavelength characterization of optical focal structures,' Opt. Commun. 145, 9-14 (1989).
    [CrossRef]
  14. S. K. Rhodes, K. A. Nugent, and A. Roberts, 'Precision measurement of the electromagnetic fields in the focal region of a high-numerical-aperture lens using a tapered fiber probe,' J. Opt. Soc. Am. A 19, 1689-1693 (2002).
    [CrossRef]
  15. M. Mansuripur, Classical Optics and its Applications (Cambridge U. Press, 2002).
  16. B. Sick, B. Hecht, and L. Novotny, 'Orientational imaging of single molecules by annular illumination,' Phys. Rev. Lett. 85, 4482-4485 (2000).
    [CrossRef] [PubMed]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, 'The focus of light--linear polarization breaks the rotational symmetry of the focal spot,' J. Mod. Opt. 50, 1917-1926 (2003).
[CrossRef]

2002 (1)

2001 (1)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

2000 (1)

B. Sick, B. Hecht, and L. Novotny, 'Orientational imaging of single molecules by annular illumination,' Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

1989 (2)

S. K. Rhodes, A. Barty, A. Roberts, and K. A. Nugent, 'Sub-wavelength characterization of optical focal structures,' Opt. Commun. 145, 9-14 (1989).
[CrossRef]

M. Mansuripur, 'Certain computational aspects of vector diffraction problems,' J. Opt. Soc. Am. A 6, 786-805 (1989).
[CrossRef]

1985 (1)

K. Tanaka, N. Saga, and K. Hauchi, 'Focusing of a Gaussian beam through a finite aperture lens,' Appl. Opt. 24, 1382-1390 (1985).
[CrossRef]

1982 (1)

1981 (1)

1977 (1)

1959 (2)

E. Wolf, 'Electromagnetic diffraction in optical systems. I. An integral representation of the image field,' Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, 'Electromagnetic diffraction in optical systems. Il. Structure of the image field in an aplanatic system,' Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Barty, A.

S. K. Rhodes, A. Barty, A. Roberts, and K. A. Nugent, 'Sub-wavelength characterization of optical focal structures,' Opt. Commun. 145, 9-14 (1989).
[CrossRef]

Belland, P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (expanded) (Cambridge U. Press, 1999).

Crenn, J. P.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, 'The focus of light--linear polarization breaks the rotational symmetry of the focal spot,' J. Mod. Opt. 50, 1917-1926 (2003).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Firester, A. H.

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Hauchi, K.

K. Tanaka, N. Saga, and K. Hauchi, 'Focusing of a Gaussian beam through a finite aperture lens,' Appl. Opt. 24, 1382-1390 (1985).
[CrossRef]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, 'Orientational imaging of single molecules by annular illumination,' Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Heller, M. E.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, 'The focus of light--linear polarization breaks the rotational symmetry of the focal spot,' J. Mod. Opt. 50, 1917-1926 (2003).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Mansuripur, M.

Marshall, G. F.

G. F. Marshall, ed., Optical Scanning (Marcel Dekker, 1991).

Novotny, L.

B. Sick, B. Hecht, and L. Novotny, 'Orientational imaging of single molecules by annular illumination,' Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Nugent, K. A.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, 'The focus of light--linear polarization breaks the rotational symmetry of the focal spot,' J. Mod. Opt. 50, 1917-1926 (2003).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Rhodes, S. K.

Richards, B.

B. Richards and E. Wolf, 'Electromagnetic diffraction in optical systems. Il. Structure of the image field in an aplanatic system,' Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Roberts, A.

Saga, N.

K. Tanaka, N. Saga, and K. Hauchi, 'Focusing of a Gaussian beam through a finite aperture lens,' Appl. Opt. 24, 1382-1390 (1985).
[CrossRef]

Schneider, M. B.

Sheng, P.

Sick, B.

B. Sick, B. Hecht, and L. Novotny, 'Orientational imaging of single molecules by annular illumination,' Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

Tanaka, K.

K. Tanaka, N. Saga, and K. Hauchi, 'Focusing of a Gaussian beam through a finite aperture lens,' Appl. Opt. 24, 1382-1390 (1985).
[CrossRef]

Webb, W. W.

Wolf, E.

B. Richards and E. Wolf, 'Electromagnetic diffraction in optical systems. Il. Structure of the image field in an aplanatic system,' Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

E. Wolf, 'Electromagnetic diffraction in optical systems. I. An integral representation of the image field,' Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (expanded) (Cambridge U. Press, 1999).

Appl. Opt. (4)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, 'The focus of light--theoretical calculation and experimental tomographic reconstruction,' Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

J. Mod. Opt. (1)

R. Dorn, S. Quabis, and G. Leuchs, 'The focus of light--linear polarization breaks the rotational symmetry of the focal spot,' J. Mod. Opt. 50, 1917-1926 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

S. K. Rhodes, A. Barty, A. Roberts, and K. A. Nugent, 'Sub-wavelength characterization of optical focal structures,' Opt. Commun. 145, 9-14 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

B. Sick, B. Hecht, and L. Novotny, 'Orientational imaging of single molecules by annular illumination,' Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (2)

E. Wolf, 'Electromagnetic diffraction in optical systems. I. An integral representation of the image field,' Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, 'Electromagnetic diffraction in optical systems. Il. Structure of the image field in an aplanatic system,' Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Other (4)

G. F. Marshall, ed., Optical Scanning (Marcel Dekker, 1991).

M. Mansuripur, Classical Optics and its Applications (Cambridge U. Press, 2002).

M. Born and E. Wolf, Principles of Optics, 7th ed. (expanded) (Cambridge U. Press, 1999).

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

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Figures (8)

Fig. 1
Fig. 1

Coordinate system for the calculation of the intensity distribution in the region of focus.

Fig. 2
Fig. 2

Error in Δ x FWHM approximated by relation (9) with respect to the exact one by Eqs. (4, 6) as a function of NA.

Fig. 3
Fig. 3

Numerical factor η ( α ) in Eq. (12).

Fig. 4
Fig. 4

Dependence of transverse and longitudinal FWHM values, Δ x FWHM and Δ z FWHM , respectively, on w 0 of the incident Gaussian beam. Vertical dotted lines indicate w 0 = R 2 and R. (a) and (b) NA = 0.4 , (c) and (d) NA = 0.75 , (e) and (f) NA = 0.95 .

Fig. 5
Fig. 5

Experimental setup for measuring the profile of the beam focused by an objective lens. Ll, L2, L3, lenses; BS, beam splitter; TS1, translation stage driven by a closed-loop feedback stepper motor; TS2, translation stage driven by closed-loop feedback PZT actuators; OL, objective lens; CCD, charge-coupled device detector; P1, P2, pinholes; C, condenser; PMT, photomultiplier tube; A1, A2, A3, scan control voltage signals from an analog–digital converter board on a personal computer. Signal A1 controls the z translation of the objective lens and signals A2 and A3 control the x and y translation of the pinhole stage. A spatial filter is formed by L1, P1, and L2.

Fig. 6
Fig. 6

Scanning electron microscope image of the pinhole ( ϕ = 0.5 ± 0.05 μ m ) used as an intensity probe in our experiment.

Fig. 7
Fig. 7

Observed x z profile in the focal region for a NA = 0.95 objective lens. The image covers a scan area of 2.5 μ m × 6 μ m .

Fig. 8
Fig. 8

Dependence of transverse ( x ) and longitudinal ( z ) FWHM values on w 0 of the incident Gaussian beam. Unconvoluted FWHMs obtained from Eq. (6) are represented by solid curves whereas the convoluted FWHMs given by Eq. (25) are drawn as dashed curves. Experimental results are marked by filled squares with error bars. Independently measured spherical aberrations were included in the calculations. Vertical dotted lines indicate w 0 = R 2 and R. (a) and (b) NA = 0.4 , (c) and (d) NA = 0.75 , (e) and (f) NA = 0.95 .

Equations (39)

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e ( r ) = i k f 2 π Ω a ( s x , s y ) s z exp { i k [ Φ ( s x , s y ) + s r ] } d s x d s y ,
a x = e 0 ( ϑ ) cos ϑ [ cos ϑ + sin 2 φ ( 1 cos ϑ ) ] ,
a y = e 0 ( ϑ ) cos ϑ ( cos ϑ 1 ) cos φ sin φ ,
a z = e 0 ( ϑ ) cos ϑ sin ϑ cos φ ,
e x ( r ) = i 2 k f ( I 0 + I 2 cos 2 ϕ ) ,
e y ( r ) = i 2 k f I 2 sin 2 ϕ ,
e z ( r ) = i k f I 1 cos ϕ ,
I 0 ( r , θ ) = 0 α e 0 ( ϑ ) cos ϑ sin ϑ ( 1 + cos ϑ ) × J 0 ( k r sin ϑ sin θ ) exp ( i k r cos ϑ cos θ ) d ϑ ,
I 1 ( r , θ ) = 0 α e 0 ( ϑ ) cos ϑ sin 2 ϑ × J 1 ( k r sin ϑ sin θ ) exp ( i k r cos ϑ cos θ ) d ϑ ,
I 2 ( r , θ ) = 0 α e 0 ( ϑ ) cos ϑ sin ϑ ( 1 cos ϑ ) × J 2 ( k r sin ϑ sin θ ) exp ( i k r cos ϑ cos θ ) d ϑ ,
e 0 ( ϑ ) = A 0 exp [ ( f sin ϑ w 0 ) 2 ]
S z ( r ) = c ( k f ) 2 32 π ( I 0 2 I 2 2 ) ,
I 0 ( r , θ = π 2 ) = A 0 0 α cos ϑ sin ϑ ( 1 + cos ϑ ) × J 0 ( k r sin ϑ ) d ϑ ,
I 1 ( r , θ = π 2 ) = A 0 0 α cos ϑ sin 2 ϑ × J 1 ( k r sin ϑ ) d ϑ ,
I 2 ( r , θ = π 2 ) = A 0 0 α cos ϑ sin ϑ ( 1 cos ϑ ) × J 2 ( k r sin ϑ ) d ϑ .
I 0 2 A 0 0 α cos ϑ sin ϑ J 0 ( k r sin ϑ ) d ϑ J 1 ( k r sin α ) k r sin α ,
Δ x FWHM 2 × 1.6163 k sin α = 0.5145 λ NA .
I 0 ( r , = z , θ = 0 ) = A 0 0 α cos ϑ sin ϑ ( 1 + cos ϑ ) × exp ( i k z cos ϑ ) d ϑ ,
I 1 ( r , θ = θ ) = I 2 ( r , θ = 0 ) = 0 .
I 0 2 A 0 0 α cos ϑ sin ϑ exp ( i k z cos ϑ ) d ϑ = 2 A 0 ( k z ) 2 k z cos α k z q exp ( i q ) d q ( sin 2 α ) [ ( sin x x ) i tan 2 α 2 ( x cos x sin x x 2 ) ] ,
Δ z FWHM = η ( α ) λ 4 sin 2 α 2 = η ( arcsin NA ) λ 4 sin 2 ( 1 2 arcsin NA ) ,
Δ z FWHM 1.772 λ 4 sin 2 α 2 = 1.772 λ 4 sin 2 ( 1 2 arcsin NA ) ,
Δ z FWHM 1.772 λ α 2 1.772 λ NA 2 ,
w 0 = f λ π w 0 ,
Δ x FWHM = 2 ln 2 w 0 0.375 λ NA eff ,
I 0 ( r , θ = π 2 ) = 0 α e 0 ( ϑ ) cos ϑ sin ϑ ( 1 + cos ϑ ) × J 0 ( k r sin ϑ ) d ϑ ,
I 1 ( r , θ = π 2 ) = 0 α e 0 ( ϑ ) cos ϑ sin 2 ϑ × J 1 ( k r sin ϑ ) d ϑ ,
I 2 ( r , θ = π 2 ) = 0 α e 0 ( ϑ ) cos ϑ sin ϑ ( 1 cos ϑ ) × J 2 ( k r sin ϑ ) d ϑ ,
I 0 2 0 α e 0 ( ϑ ) ϑ J 0 ( k r ϑ ) d ϑ ,
I 1 0 α e 0 ( ϑ ) ϑ 2 J 1 ( k r ϑ ) d ϑ ,
I 2 1 2 0 α e 0 ( ϑ ) ϑ 3 J 2 ( k r ϑ ) d ϑ .
I 0 0 α exp [ ( f ϑ w 0 ) 2 ] ϑ J 0 ( k r ϑ ) d ϑ 0 f α w 0 exp ( x 2 ) x J 0 ( k r w 0 f x ) d x 0 α exp ( x 2 ) x J 0 ( ρ x ) d x = exp [ ( ρ 2 ) 2 ] ,
z 0 = π w 0 2 λ = λ π ( f w 0 ) 2 .
Δ z FWHM = 2 ( λ π ) ( f w 0 ) 2 0.6366 λ NA eff 2 .
I 0 ( r , θ = 0 ) = 0 α e 0 ( ϑ ) cos ϑ sin ϑ ( 1 + cos ϑ ) × exp ( i k r cos ϑ ) d ϑ ,
I 1 ( r , θ = 0 ) = 0 = I 2 ( r , θ = 0 ) .
I 0 0 exp ( x 2 ) x exp { i k r [ 1 1 2 ( w 0 x f ) 2 ] } d x = 1 2 exp ( i k r ) 0 exp ( q ) exp [ i 2 k r ( w 0 f ) 2 q ] d q ( 1 + i k r w 0 2 2 f 2 ) 1 ,
I 0 2 1 r 2 + ( 2 f 2 k w 0 2 ) 2 ,
S ̃ z ( x , y ) = S z ( x , y ) P ( x x , y y ) d x d y ,

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